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Stable non-Gaussian random processes: stochastic models with infinite variance. (English) Zbl 0925.60027
Stochastic Modeling. New York, NY: Chapman & Hall. xviii, 632 p. (1994).
This is an encyclopaedic monograph on non-Gaussian stable vectors, stable processes and stochastic integrals with respect to stable processes. The authors convince us that the class of stable processes with infinite variance is as rich and interesting as the class of Gaussian processes. They treat many different types of stable processes, such as fractional stable processes, stable moving averages, stable harmonisable processes, the stable Ornstein-Uhlenbeck process, and stationary stable processes. The reader is introduced into a new world of processes with large jumps and wild oscillations. Because of this erratic behaviour, stable processes with infinite variance seem appropriate models for real phenomena with heavy-tailed distributions, such as crashes and catastrophes. Therefore they have attracted the attention of many applied workers in meteorology, economics, finance, insurance, physics, etc.
There is a vast amount of mathematical work on stable processes. The book under review is the first comprehensive treatment of the topic. It collects a wealth of results and applications, including the substantial contributions from the research of the authors. In the first chapter the basics of stable random variables are given. Both the classical definition via characteristic functions and the representation via series are provided. The second chapter deals with multivariate stable vectors. Analogues to orthogonality for infinite variance variables are discussed. Chapter 3 concerns stable random processes and stochastic integrals. Stable stochastic integrals are treated in great detail, and several options for definitions are discussed. In the fourth chapter the dependence structure of stable vectors is considered, including problems on linear regression, linear dependence, joint moments, association, and pseudo-orthogonality. Chapter 5 deals with nonlinear regression. Chapter 6 treats the important issue of complex stable stochastic integrals and harmonisable stable processes.
Chapter 7 deals with more general stochastic processes: self-similar processes. This is a very good rigorous mathematical treatment of the topic, including the Gaussian case. Fractional Brownian motion and numerous examples of stable self-similar processes are studied. Chapter 8 is on Chentsov random fields. Chapters 9-13 deal with various aspects of stable processes: boundedness, continuity, oscillations, zero-one laws, measurability, integrability. Chapter 14 gives very valuable historical remarks on the results in the different chapters and contains an overview of current research in various fields touched on in the book. It also includes many references to unpublished papers. An appendix contains tables of symmetric stable fractiles. They show the great differences between the infinite and finite variance (= Gaussian) stable distributions. The bibliography of 16 pages gives a complete picture of the field. Extensive subject and author indices are provided at the end of the book. Every chapter contains computer graphics illustrating the erratic behaviour of stable processes and their differences from Gaussian processes. At the end of each chapter one can find exercises which make this book attractive for seminars and graduate courses at universities.
It is primarily a book for the researcher in stochastic processes and its applications, but it is written in a style which makes it accessible also to the student or applied worker as an introduction and reference book on stable and self-similar processes. In summary, this is an excellent book on stable processes. It will become the standard reference on the topic.

60G05 Foundations of stochastic processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60E07 Infinitely divisible distributions; stable distributions
60G18 Self-similar stochastic processes
60G99 Stochastic processes
60H99 Stochastic analysis
62J05 Linear regression; mixed models
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)